The smallest boat moored at the Hemingway Marina in Havana may hold the most intrigue .

Hunkered down inside a blue , 25-foot sailboat named Salty are Josh Hakken and his wife , Sharyn , and their two boys , 2-year-old Chase and 4-year-old Cole .

Theirs is no ordinary visit to the historic port , where they have eluded capture but where CNN found them Tuesday . The Hakkens have been on the lam after they allegedly snatched the two boys from their grandmother 's home in Florida .

The couple lost custody of their children last year . There is an international manhunt for this family , and here they are , blending in among the other boats .

But their stop appears likely to be temporary , as Cuban officials announced Tuesday afternoon that they plan to turn the family over to U.S. authorities .

Josh Hakken glared through his sunglasses at the CNN reporter who found him and said nothing beyond confirming his identity .

The two boys are OK , said a woman matching the description of his wife , and she left it at that .

Josh and Sharyn Hakken are wanted by U.S. authorities and were not in custody in Cuba . But that does n't mean they are n't being watched .

Cuban security officials wearing sidearms appeared as the CNN video crew was filming and ordered them to stop . At the security guards ' request , CNN stepped away from the boat , which looked just like the photo that Florida law enforcement officials had circulated , except more battered .

A statement from the Cuban foreign ministry said the boat put in to the marina , located a few miles west of Havana , in bad weather on Sunday .

`` From the first moment , diplomatic notes were exchanged and a permanent and professional communication has been maintained between MINREX '' -- Cuba 's foreign ministry -- `` and the U.S. Interests Section in Havana , with the goal of guaranteeing the integrity and well-being of the minors , '' the statement said .

Joshua Hakken : Libertarian washed ashore in the worker 's paradise

U.S. government officials have told the sheriff 's office in Hillsborough County , Florida -- which includes the city of Tampa from which the boys went missing -- that `` they are receiving exceptional cooperation from the Cuban government , '' the sheriff 's office said Tuesday afternoon .

Earlier in the day , U.S. State Department spokesman Patrick Ventrell said the State Department is aware of the case , though it could not provide additional information because of privacy concerns .

`` But what I do want to say , more broadly speaking , is that one of the department 's highest priorities is the welfare of U.S. citizens overseas , and this is particularly true for children who are our most vulnerable citizens , '' Ventrell said .

Some worried that recovering the children will be difficult .

`` Unfortunately , these parents and these poor children , these innocent ones , will now be in a country where there are no laws , there is no redress , and that has been a refuge for fugitives and wanted criminals for many years , '' Rep. Ileana Ros-Lehtinen , R - Florida , told CNN .

The boys had been removed from the care of the Florida couple last year , and on April 2 , the couple 's parental rights were terminated in Louisiana , investigators say .

The Hillsborough County sheriff 's office described Josh Hakken as an anti-government protester , and he was believed to be armed , according to the Hillsborough County sheriff 's office .

There is no extradition treaty between Cuba and the United States , although there have been recent cases in which Americans sought for crimes in the United States and discovered in Havana have been sent back by the Cubans .

The FBI currently estimates there are around 70 fugitives from U.S. justice in Cuba , which is one of the reasons Cuba remains on the U.S. list of countries that support state terrorism . Most of the American fugitives in Cuba have been there for decades and have ties to revolutionary movements or radical groups .

Last week , the police department in Slidell , Louisiana , issued its own statement offering background on the Hakkens and why the boys were taken from the parents last year .

In June of 2012 , Slidell police responded to a disturbance report at a hotel where Josh and Sharyn Hakken were staying with their sons , the police statement said .

`` When police arrived , both Mr. and Mrs. Hakken were acting in a bizarre manner that alarmed officers . They were talking about ` completing their ultimate journey ' and were traveling across the country to ` take a journey to the Armageddon ' , '' the Slidell police statement said , adding , `` Let it be noted that both of their children were present in the hotel room at the time . ''

Because of the parents ' behavior and `` the fact that narcotics and weapons were located inside of the hotel room , '' the children were taken by child welfare officers , and Joshua Hakken was arrested on drug charges , the statement said .

`` Approximately two weeks later , Slidell Police were notified that Mr. Hakken had shown up to the foster family home ... with a firearm demanding the return of his children , '' the Slidell police statement continued . `` The foster parents called 911 , and Mr. Hakken fled without his children . We have heard nothing until -LRB- Wednesday -RRB- . ''

At some point over the past few months , the children were sent to live with their grandmother , Patricia Hauser , the mother of Sharyn Hakken .

Sheriff 's investigators say Josh Hakken entered Hauser 's home at 6:30 a.m. last Wednesday . She told police that he tied her up and fled with the children in her silver 2009 Toyota Camry . That vehicle was found later that day just a couple of blocks away from the home .

Those investigators told CNN they believe Hakken joined up with his wife , who was waiting in their pickup truck , and the family drove to a parking garage . A short time later , investigators said , Hakken is believed to have taken a sailboat out of a private slip in nearby Madeira Beach .

Surveillance images showed the boat sailing into the Gulf of Mexico about three and a half hours after the boys disappeared from their grandmother 's home , investigators said , adding that the photos showed adults and children on board .

@highlight

FIRST ON CNN : A man on a boat in Havana tells CNN he is Josh Hakken

@highlight

Cuba says it will turn the Hakkens back over to the United States

@highlight

Authorities believe Hakken abducted his two small sons last week

@highlight

The boys were taken from their parents by child welfare officials in Louisiana last year